§25.14 Lerch’s Transcendent ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions (original) (raw)

Contents
  1. §25.14(i) Definition
  2. §25.14(ii) Properties

§25.14(i) Definition

25.14.1 Φ⁡(z,s,a)≡∑n=0∞zn(a+n)s,
|z <1; ℜ⁡s>1,
ⓘ Defines: Φ⁡(z,s,a): Lerch’s transcendent Symbols: ≡: equals by definition,ℜ⁡: real part,n: nonnegative integer,a: real or complex parameter,s: complex variable andz: complex variable Keywords: definition, infinite series, series representation Source: Erdélyi et al. (1953a, (1.11.1), p. 27) Referenced by: (25.14.2),(25.14.3),§25.14(ii),§25.14,Erratum (V1.0.21) for Equation (25.14.1) Permalink: http://dlmf.nist.gov/25.14.E1 Encodings: TeX, pMML, png Clarification (effective with 1.0.21): The previous constraint a≠0,−1,−2,…, was removed. A clarification regarding the correct constraints for Lerch’s transcendentΦ⁡(z,s,a) has been added in the text immediately below. See also: Annotations for §25.14(i),§25.14 andCh.25

If s is not an integer then |ph⁡a|<π; if s is a positive integer then a≠0,−1,−2,…; if s is a non-positive integer then a can be any complex number. For other values of z, Φ⁡(z,s,a) is defined by analytic continuation. This is the notation used in Erdélyi et al. (1953a, p. 27).Lerch (1887) used𝔎⁡(a,x,s)=Φ⁡(e2⁢π⁢i⁢x,s,a).

The Hurwitz zeta function ζ⁡(s,a) (§25.11) and the polylogarithm Lis⁡(z) (§25.12(ii)) are special cases:

25.14.2 ζ⁡(s,a)=Φ⁡(1,s,a),
ℜ⁡s>1, a≠0,−1,−2,…,
ⓘ Symbols: ζ⁡(s,a): Hurwitz zeta function,Φ⁡(z,s,a): Lerch’s transcendent,ℜ⁡: real part,a: real or complex parameter ands: complex variable Keywords: specialization Proof sketch: Derivable from (25.11.1), (25.14.1). Permalink: http://dlmf.nist.gov/25.14.E2 Encodings: TeX, pMML, png See also: Annotations for §25.14(i),§25.14 andCh.25
25.14.3 Lis⁡(z)=z⁢Φ⁡(z,s,1),
ℜ⁡s>1, |z ≤1.
ⓘ Symbols: Φ⁡(z,s,a): Lerch’s transcendent,Lis⁡(z): polylogarithm,ℜ⁡: real part,s: complex variable andz: complex variable Keywords: specialization Proof sketch: Derivable from (25.12.10) and (25.14.1). Referenced by: (25.12.12) Permalink: http://dlmf.nist.gov/25.14.E3 Encodings: TeX, pMML, png See also: Annotations for §25.14(i),§25.14 andCh.25
25.14.3_1 za⁢Φ⁡(z,s,a)=Γ⁡(1−s)⁢(−ln⁡z)s−1+∑n=0∞ζ⁡(s−n,a)⁢(ln⁡z)nn!,
|ln⁡z <2⁢π; s≠1,2,3,…, a≠0,−1,−2,….
ⓘ Symbols: Γ⁡(z): gamma function,ζ⁡(s,a): Hurwitz zeta function,Φ⁡(z,s,a): Lerch’s transcendent,π: the ratio of the circumference of a circle to its diameter,!: factorial (as in n!),ln⁡z: principal branch of logarithm function,n: nonnegative integer,a: real or complex parameter,s: complex variable andz: complex variable Source: Erdélyi et al. (1953a, (1.11.8), p. 29) Referenced by: §25.14(i),Erratum (V1.2.1) for §25.14(i) Permalink: http://dlmf.nist.gov/25.14.E3_1 Encodings: TeX, pMML, png Addition (effective with 1.2.1): This equation was added. See also: Annotations for §25.14(i),§25.14 andCh.25

If s=m a positive integer then

25.14.3_2 za⁢Φ⁡(z,m,a)=∑′n=0′∞′⁢ζ⁡(m−n,a)⁢(ln⁡z)nn!+(ln⁡z)m−1(m−1)!⁢(ψ⁡(m)−ψ⁡(a)−ln⁡(−ln⁡z)),
|ln⁡z <2⁢π; m=2,3,4,…, a≠0,−1,−2,….
ⓘ Symbols: ζ⁡(s,a): Hurwitz zeta function,Φ⁡(z,s,a): Lerch’s transcendent,π: the ratio of the circumference of a circle to its diameter,ψ⁡(z): psi (or digamma) function,!: factorial (as in n!),ln⁡z: principal branch of logarithm function,m: nonnegative integer,n: nonnegative integer,a: real or complex parameter andz: complex variable Source: Erdélyi et al. (1953a, (1.11.9), p. 30) Referenced by: §25.14(i),Erratum (V1.2.1) for §25.14(i) Permalink: http://dlmf.nist.gov/25.14.E3_2 Encodings: TeX, pMML, png Addition (effective with 1.2.1): This equation was added. See also: Annotations for §25.14(i),§25.14 andCh.25

Here the prime signifies that the term for n=m−1 is to be omitted. In the case s=1 we have

25.14.3_3 a⁢Φ⁡(z,1,a)=F⁡(a,1;a+1;z),
|z <1.
ⓘ Symbols: F⁡(a,b;c;z) or F⁡(a,bc;z): =F12⁡(a,b;c;z)Gauss’ hypergeometric function,Φ⁡(z,s,a): Lerch’s transcendent,a: real or complex parameter andz: complex variable Source: Erdélyi et al. (1953a, (1.11.10), p. 30) Referenced by: §25.14(i),Erratum (V1.2.1) for §25.14(i) Permalink: http://dlmf.nist.gov/25.14.E3_3 Encodings: TeX, pMML, png Addition (effective with 1.2.1): This equation was added. See also: Annotations for §25.14(i),§25.14 andCh.25

For hypergeometric function F see 15.2(i).

§25.14(ii) Properties

With the conditions of (25.14.1) and m=1,2,3,…,

25.14.4 Φ⁡(z,s,a)=zm⁢Φ⁡(z,s,a+m)+∑n=0m−1zn(a+n)s.
25.14.5 Φ⁡(z,s,a)=1Γ⁡(s)⁢∫0∞xs−1⁢e−a⁢x1−z⁢e−x⁢dx,
ℜ⁡s>1, ℜ⁡a>0 if z=1;ℜ⁡s>0, ℜ⁡a>0 if z∈ℂ∖[1,∞).
ⓘ Symbols: Γ⁡(z): gamma function,Φ⁡(z,s,a): Lerch’s transcendent,[a,b): half-closed interval,ℂ: complex plane,dx: differential of x,∈: element of,e: base of natural logarithm,∫: integral,ℜ⁡: real part,∖: set subtraction,x: real variable,a: real or complex parameter,s: complex variable andz: complex variable Keywords: improper integral, integral representation Source: Erdélyi et al. (1953a, (1.11.3), p. 27) Referenced by: Erratum (V1.1.4) for Equation (25.14.5) Permalink: http://dlmf.nist.gov/25.14.E5 Encodings: TeX, pMML, png See also: Annotations for §25.14(ii),§25.14 andCh.25
25.14.6 Φ⁡(z,s,a)=12⁢a−s+∫0∞zx(a+x)s⁢dx−2⁢∫0∞sin⁡(x⁢ln⁡z−s⁢arctan⁡(x/a))(a2+x2)s/2⁢(e2⁢π⁢x−1)⁢dx,
ℜ⁡a>0 if |z <1;ℜ⁡s>1, ℜ⁡a>0 if z =1.
ⓘ Symbols: Φ⁡(z,s,a): Lerch’s transcendent,π: the ratio of the circumference of a circle to its diameter,dx: differential of x,e: base of natural logarithm,∫: integral,arctan⁡z: arctangent function,ln⁡z: principal branch of logarithm function,ℜ⁡: real part,sin⁡z: sine function,x: real variable,a: real or complex parameter,s: complex variable andz: complex variable Keywords: improper integral, integral representation Source: Erdélyi et al. (1953a, (1.11.4), p. 28) Notes: For the case |z <1 see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case z =1 one checks that the first integral converges absolutely iff ℜ⁡s>1. Referenced by: Erratum (V1.1.4) for Equation (25.14.6) Permalink: http://dlmf.nist.gov/25.14.E6 Encodings: TeX, pMML, png Clarification (effective with 1.1.4): The constraint which originally read “ℜ⁡s>0 if z <1; ℜ⁡s>1 if

Lerch’s transformation formula

25.14.7 Φ⁡(z,s,a)=i⁢(2⁢π)s−1za⁢Γ⁡(1−s)⁢(e−π⁢i⁢s/2⁢Φ⁡(e−2⁢π⁢i⁢a,1−s,ln⁡z2⁢π⁢i)−eπ⁢i⁢(2⁢a+(s/2))⁢Φ⁡(e2⁢π⁢i⁢a,1−s,1−ln⁡z2⁢π⁢i)).

For these and further properties see Erdélyi et al. (1953a, pp. 27–31).

25.14.8 Φ⁡(−z,s,a)=12⁢π⁢i⁢∫σ−i⁢∞σ+i⁢∞Γ⁡(1+t)⁢Γ⁡(−t)⁢zt(a+t)s⁢dt,
|arg⁡z <π, ℜ⁡a>0,
ⓘ Symbols: Γ⁡(z): gamma function,Φ⁡(z,s,a): Lerch’s transcendent,π: the ratio of the circumference of a circle to its diameter,dx: differential of x,i: imaginary unit,∫: integral,ℜ⁡: real part,a: real or complex parameter,s: complex variable andz: complex variable Source: Olde Daalhuis (2024, (1.5)) Referenced by: §25.14(ii),Erratum (V1.2.1) for §25.14(ii) Permalink: http://dlmf.nist.gov/25.14.E8 Encodings: TeX, pMML, png Addition (effective with 1.2.1): This equation was added. See also: Annotations for §25.14(ii),§25.14 andCh.25

with max⁡(−ℜ⁡a,−1)<σ<0. This Mellin–Barnes integral representation is used in Olde Daalhuis (2024) to obtain large |z| asymptotic approximations forΦ⁡(−z,s,a). In the special case s=m an integer these asymptotic approximations simplify

25.14.9 Φ⁡(−z,m,a)=−πza⁢∑n=0m−1bn⁢(ln⁡z)m−n−1Γ⁡(m−n)−∑n=1∞(−z)−n(a−n)m,
|arg⁡z <π.
ⓘ Symbols: Γ⁡(z): gamma function,Φ⁡(z,s,a): Lerch’s transcendent,π: the ratio of the circumference of a circle to its diameter,ln⁡z: principal branch of logarithm function,m: nonnegative integer,n: nonnegative integer,a: real or complex parameter andz: complex variable Source: Olde Daalhuis (2024, (1.6)) Referenced by: §25.14(ii),Erratum (V1.2.1) for §25.14(ii) Permalink: http://dlmf.nist.gov/25.14.E9 Encodings: TeX, pMML, png Addition (effective with 1.2.1): This equation was added. See also: Annotations for §25.14(ii),§25.14 andCh.25

The first sum is zero in the case that m is a non-positive integer. In the case that m is a positive integer we have the additional constraint a≠1,2,3,…. The coefficients bn are the Taylor coefficients of csc⁡(π⁢(t−a)) about t=0.

The small and large a asymptotics is discussed in Cai and López (2019), Ferreira and López (2004), Paris (2016), and the asymptotics as ℜ⁡s→−∞ is discussed in Navas et al. (2013).